4.9 Growth kinetics

So far we have described the basic kinetics of mycelial growth in words. Converting them to algebra results in the relationships we illustrated in Figs 7-9 being expressed in the equation:

Ē = µmaxG

where Ē is the mean extension rate of the colony margin;µmax is the maximum specific (biomass) growth rate;
and G is the hyphal growth unit length.

As can be seen from Fig. 13 and Table 2 , the factors which determine the radial growth rate of the colony (Kr) are the specific growth rate of the fungus (μ) and the width of the peripheral growth zone (w); that is, Kr = wμ.

The fungal colony therefore grows outward radially at a linear rate (that is, an arithmetic plot of colony radius against time forms a straight line), continually growing into unexploited substratum. As it does so, the production of new branches ensures the efficient colonisation and utilisation of the substratum. For the colony as a whole, the peripheral growth zone is a ring of active tissue at the colony margin which is responsible for expansion of the colony. At the level of the individual hypha, the peripheral growth zone corresponds to the volume of hypha contributing to extension growth of the apex of that hypha (the hyphal growth unit).

The rate of change in conditions below a colony will be related to the density (biomass per unit surface area) of the fungal biomass supported. It follows from this that a profusely branching mycelium (low value of G) will develop unfavourable conditions in the medium below the colony more rapidly than a sparsely branching mycelium (high value of G). Consequently, a relationship between G and w would be anticipated, and is observed. It means, for example, that Kr can be used to study the effect of temperature on fungal growth because w is not affected appreciably by temperature, however the concentration of glucose (for example) does affect w, so Kr cannot be used to investigate the effect of nutrient concentration. The biological consequence of this is that filamentous fungi can maintain maximal radial growth rate over nutrient-depleted substrates.

Unlike colonies formed by unicellular bacteria and yeasts, where colony
expansion is the result of the production of daughter cells and occurs only
slowly, the ability of filamentous fungi to direct all their growth capacity
to the hyphal apex allows the colony to expand far more rapidly.
Importantly, the fungal colony expands at a rate which exceeds the rate of
diffusion of nutrients from the surrounding substratum.

Although nutrients under the colony are rapidly exhausted, the hyphae at the edge of the colony have only a minor effect on the substrate concentration and continue to grow outwards, exploring for more nutrients. In contrast, the rate of expansion of bacterial and yeast colonies is extremely slow and less than the rate of diffusion of nutrients (Table 3). Colonies of unicellular organisms quickly become diffusion limited and therefore, unlike fungal colonies, can only attain a finite size.

We have discussed how hyphal extension growth follows a few general
relationships that are conveyed in relatively simple equations (above and
Section 4.4); so, it follows that hyphal growth kinetics are well suited to
mathematical modelling, using these word-equations:

Ē, the mean tip extension rate, is given by µmax (the maximum specific
growth rate) multiplied by G, the hyphal growth unit.

G is defined as the average length of a hypha supporting a
growing tip.

G, consequently, is given by
Lt, total mycelial length, divided by Nt, the total number of tips.

In a
fungal colony, the hyphal growth unit is approximately equal to the width of
the peripheral growth zone, which is a ring-shaped peripheral area of the
mycelium that contributes to radial expansion of the colony.

In a mycelium that is exploring the substrate, branching will be
rare and so G will be large.G is therefore an indicator of branching density.

A new
branch is initiated when the capacity for a hypha to extend increases above
Ē, thereby regulating G to a uniform value indicative of the characteristic
branching density of that fungus under those growing conditions.

All
these features of normal filamentous hyphal growth can be expressed
algebraically in a vector-based mathematical model in which the growth
vector of each virtual hyphal tip is calculated at each iteration of the
algorithm by reference to the surrounding virtual mycelium. For example, the
Neighbour-Sensingcomputer program starts with a single hyphal tip,
equivalent to a fungal spore. Each time the program runs through its
algorithm the tip advances by a growth vector (initially set by the user)
and may branch (with an initial probability set by the user).

In the
Neighbour-Sensing program each hyphal tip is an active agent, described by
its three-dimensional position in space, length, and growth vector, that can
vector within three-dimensional data space using rules of exploration that
are set (initially by the experimenter) within the program. The rules are
biological characteristics such as:

the basic kinetics of in vivo hyphal
growth,

branching characteristics (frequency, angle, position),

tropic
field settings that involve interaction with the environment.

The
experimenter can alter parameters to investigate their effect on form; the
final geometry is reached by the program (not the experimenter) adapting the
biological characteristics of the active agents during their growth, as in
life. This is called the Neighbour-Sensing model and it brings together the
essentials of hyphal growth kinetics into mathematical cyberfungus that can
be used for experimentation on the theoretical rules governing hyphal
patterning and tissue morphogenesis (Meškauskas et al., 2004 a & b).

The
Neighbour-Sensing model ‘grows’ a simulated cybermycelium using realistic
branching rules decided by the user. As the cyberhyphal tips grow out into
the modelling space the model tracks where they have been, and those tracks
become the hyphal threads of the cybermycelium. All positioning information
is stored by the model as numerical data and so the data handling work
becomes more and more extensive as branching produces more hyphal tips and
the cybermycelium ‘grows’ in three dimensions on the computer monitor; it is
this steady growth process that generates the very large amount of data.

The process of simulation is programmed as a closed loop. This loop is
performed for each currently existing hyphal tip of the mycelium and the
algorithm:

Finds the number of neighbouring segments of mycelium (N). A
segment is counted as neighbouring if it is closer than the given critical
distance (R). In the simplest case we did not use the concept of the density
field, preferring a more general formulation about the number of the
neighbouring tips.

If N<Nbranch (the given number of neighbours required to suppress
branching), there is a certain given probability (Pbranch) that the tip
will branch. If the generated random number (0...1) is less than this
probability, the new branch is created, and the branching angle takes a
random value. The location of the new tip initially coincides with the
current tip. This stochastic branch generation model is similar overall
to earlier ones in which distance between branches and branching angles
followed experimentally measured statistical distributions.

Initial
versions of the model did not implement tropic reactions (to test the kind
of morphogenesis that might arise without this component). Later versions of
the model tested how autotropic reactions affected the simulation. This
model is predictive and successfully describes the growth of hyphae, so
confirming its credibility and indicating plausible links between the
equations and real physiology; but it is just one of several mathematical
models of fungal growth that have been published. For a wider view of this
research we refer you (in
alphabetical order) to Bartnicki-Garcia et al. (1989), Boswell et al.
(2003), Boswell (2008), Davidson (2007), Goriely & Tabor (2008), Moore
et
al. (2006), Moore & Meškauskas (2017), Prosser (1990,
1995a & b)and Vidal-Diez de
Ulzurrun et al. (2015).

Most models published so far simulate growth of mycelia on a
two-dimensional plane; the Neighbour-Sensing model, however, whilst being as
simple as possible, is able to simulate formation of a spherical, uniformly
dense fungal colony in a visualisation in three-dimensional space. A
description of the mathematics on which the model is based can be found in
Moore et al. (2006); we will not dwell on this aspect here. The complete
application can be downloaded for personal experimentationelsewhere on our host website [at this URL:
http://www.davidmoore.org.uk/CyberWEB/index.htm].

The
Neighbour-Sensing model successfully imitates the three branching strategies
of fungal mycelia illustrated by Nils Fries in 1943 (Fig.
16 compares
computer simulations with the original 1943 illustrations shown previously
in Figs 3 to 5).

Fig. 16. Simulation of the three different
colony types described by Fries (1943). Panel A shows the
Boletus type, B the Amanita type and C the
Tricholoma type. The modelling parameters used for each of these
simulations are described in the text. The simulation is the upper figure in
each case.

The Neighbour-Sensing model shows that
random growth and branching (i.e. a model that does not include the local
hyphal tip density field effect or any other tropism) is sufficient to form
a spherical colony. The colony formed by such a model is more densely
branched in the centre and sparser at the border; a feature observed in
living mycelia. Models incorporating local hyphal tip density field to
affect patterning produced the most regular spherical colonies. As with the
random growth models, making branching sensitive to the number of
neighbouring tips forms a colony in which a near uniformly dense,
essentially spherical, core is surrounded by a thin layer of slightly less
dense mycelia.

Using the branching types discussed by Fries (1943) as a
comparison, the morphology of virtual colonies produced when branching (but
not growth vector) was made sensitive to the number of neighbouring tips was
closest to the so-called Boletus type (Fig
16A). This suggests that
the Boletus type branching strategy does not use tropic reactions to
determine patterning, nor some pre-defined branching algorithm. Evidently,
hyphal tropisms are not always required to explain ‘circular’ mycelia (that
is, mycelia that are spherical in three-dimensions).

When the
Neighbour-Sensing model implements the negative autotropism of hyphae, a
spherical, near uniformly dense colony is also formed, but the structure
differs from the previously mentioned Boletus type, being more similar to
the Amanita rubescens type, characterised by a certain degree of
differentiation between hyphae (Fig. 16B):

first rank hyphae
tending to grow away from the centre of the colony;

second rank hyphae
growing less regularly and filling the remaining space.

In the early
stages of development such a colony is more star-like than spherical. It is
worth emphasising that this remarkable differentiation of hyphae emerges in
the visualisation even though all virtual hyphae are driven by the same
algorithm. The program does not include routines implementing differences in
hyphal behaviour.

Finally, when both autotropic reaction and branching
are regulated by the hyphal density field, a spherical, uniformly dense
colony is also formed. However, the structure is different again, such a
colony being like the Tricholoma type illustrated by Fries (1943) (Fig.
16C). This type has the appearance of a dichotomous branching pattern,
but it is not a true dichotomy. Rather the new branch, being very close,
generates a strong density field that turns the growth vector of the older
tip away from the new branch.

Hence the Amanita rubescens and Tricholoma
branching strategies may be based on a negative autotropic reaction of the
growing hyphae while the Boletus strategy may be based on the absence of
such a reaction, relying only on density-dependent branching. Differences
between Amanita and Tricholoma in the way that the growing tip senses its
neighbours may be obscured in life. In Amanita and Boletus types, the tip
may sense the number of other tips in its immediate surroundings. In the
Tricholoma type, the tip may sense all other parts of the mycelium, but the
local segments have the greatest impact.

This model shows that the
broadly different types of branching observed in the fungal mycelium are
likely to be based on differential expression of relatively simple control
mechanisms. The ‘rules’ governing branch patterning (that is, the mechanisms
causing the patterning) are likely to change in the life of a mycelium, as
both intracellular and extracellular conditions alter. Some of these changes
can be imitated by making alterations to specific model parameters during a
simulation. By switching between parameter sets it is possible to produce
more complex structures.

Experiments with the model simulated both
colonial growth of the sort that occurs in Petri dish cultures (Fig.
17),
and development of a mushroom-shaped ‘fruit body’ (Fig.
18). These
experiments make it evident that it is not necessary to impose complex
spatial controls over development of the mycelium to achieve specific
geometrical forms. Rather, geometrical form of the mycelium emerges because
of the operation of specific locally-effective hyphal tip interactions.

Fig.17. Simulation
of colonial growth of the sort that occurs in Petri dish cultures. Oblique
view (top) and slice of the colony (bottom), where secondary branching was
activated at the 220-time unit. The secondary branches had negative
gravitropism. For both primary and secondary branches the growth was
simulated assuming negative autotropic reaction and density-dependent
branching. If the density allowed branching, the branching probability was
40% per iteration (per time unit). The final age of the colony was 294-time
units. Secondary branches are colour-coded red, and hyphae of the primary
mycelium are coloured green (oldest) to magenta (youngest), depending on the
distance of the hyphal segment from the centre of the colony (modified from
Meškauskas et al., 2004b; reproduced with
permission from Elsevier).

Fig. 18. Simulation of a (3-dimensional) mushroom primordium. A spherical colony was first
'grown' for 76-time
units. This was converted into an organised structure, like the developing
mushroom stem by applying the parallel galvanotropism for 250-time units.
Subsequent application of a positive gravitropic reaction formed a cap-like
structure (1,000-time units) (modified from Meškauskas et al., 2004a;
reproduced with permission from Elsevier).

These computer simulations suggest
that because of the kinetics of hyphal tip growth, very little regulation of
cell-to-cell interaction is required to generate the overall architecture of
fungal fruit body structures or the basic patterning of the mycelium.
Specifically:

Complex fungal fruit body shapes can be simulated by
applying the same regulatory functions to all the growth points active in a
structure at any specific time.

The shape of the fruit body emerges as
the entire population of hyphal tips respond together, in the same way, to
the same signals.

No global control of fruit body geometry is necessary
(Meškauskas et al., 2004a).

The experiments described above have exposed a
fascinating feature of the ‘crowd behaviour’ of fungal hyphal tips, which is
that the shapes of complex fungal fruit bodies can be simulated by applying
the same regulatory functions to every one of the growth points active in a
structure at any specific time. All parameter sets that generate shapes
reminiscent of fungal fruit bodies feature a sequence of changes in
parameter settings that are applied to all hyphal tips in the simulation. No
localised regulation is necessary. Absence of global control of fruit body
geometry does not necessarily imply an absence of localised control of
details of fruit body structure. Indeed, by its very nature, the ‘sensing of
neighbouring hyphae’ aspects of the model would support the interpretation
of ‘Reijnders’ hyphal knots’ (Reijnders, 1963; and see Chapter 12,
especially Section 12.16) as a central ‘inducer’ hypha organising
differentiation of a small group of surrounding hyphae to regulate detailed
structures within fruit body tissues.

The remarkable reality of the
simulations generated by the Neighbour-Sensing program encourages confidence
in the accuracy and reliability of the Neighbour-Sensing mathematical model
on which it is based. That confidence leads us to believe that the model is
revealing unexpected capabilities of the hyphal lifestyle of fungi; but we
feel that we have only just scratched the surface of what this mathematical
model is able to reveal; the model is not yet perfect. A feature that
remains to be implemented in the model is hyphal fusion (or hyphal
anastomosis), which is such an important feature of living mycelia (Section
5.16). Initial work on the mathematics of this suggests that hyphal fusion
can be catered for in the algorithms underlying the Neighbour-Sensing model.
Inclusion of anastomosis would enable the model to generate biologically
inspired networks and so provide a tool to analyse these networks to yield
information about connectivity, minimum path length, etc. In addition, it
would be possible to address network robustness in silico by investigating
the effect of removal of network links on connectivity.

Looking further
into the future, it should be possible to add physiological data, such as
substrate uptake and substrate transport kinetics, to growth and branching.
Since the Neighbour-Sensing model ‘grows’ a realistic mycelium and tracks
all the hyphal segments so generated, there is no mathematical impediment to
assigning to those hyphal segments the algebraic characteristics defined to
describe substrate uptake, utilisation and translocation kinetics, and their
variation with age of the hyphal section.

The opportunity to tailor
parameter sets (or ‘strategies’) to simulate specific species of fungi (the
individual parameter sets being our cyberspecies) became evident in our
first experiments with the model in which distinctions could be made between
cybermycelia with morphological similarities to the Boletus, Amanita and
Tricholoma types of young mycelia discussed above. Comparable microscopic
observations of young mycelia of any live fungus should enable the
derivation of parameter sets that produce cybermycelia which are exact
simulations of the living material.

This may also contribute to
understanding hyphal and mycelial evolution because we might imagine that
the evolutionary origins of specific aspects of the kinetics of hyphal
growth and branching could be revealed by comparison of cyberspecies
representing living taxa with known evolutionary relationships.